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<title>module Jacobian - bigdecimal: Ruby Standard Library Documentation</title>


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  <h3>Methods</h3>

  <ul class="link-list" role="directory">
    <li ><a href="#method-i-dfdxi">#dfdxi</a>
    <li ><a href="#method-i-isEqual">#isEqual</a>
    <li ><a href="#method-i-jacobian">#jacobian</a>
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    <main role="main" aria-labelledby="module-Jacobian">
    <h1 id="module-Jacobian" class="module">
      module Jacobian
    </h1>

    <section class="description">
    
<p>require ‘bigdecimal/jacobian’</p>

<p>Provides methods to compute the <a href="Jacobian.html"><code>Jacobian</code></a> matrix of a set of equations at a point x. In the methods below:</p>

<p>f is an Object which is used to compute the <a href="Jacobian.html"><code>Jacobian</code></a> matrix of the equations. It must provide the following methods:</p>
<dl class="rdoc-list note-list"><dt>f.values(x)
<dd>
<p>returns the values of all functions at x</p>
</dd><dt>f.zero
<dd>
<p>returns 0.0</p>
</dd><dt>f.one
<dd>
<p>returns 1.0</p>
</dd><dt>f.two
<dd>
<p>returns 2.0</p>
</dd><dt>f.ten
<dd>
<p>returns 10.0</p>
</dd><dt>f.eps
<dd>
<p>returns the convergence criterion (epsilon value) used to determine whether two values are considered equal. If |a-b| &lt; epsilon, the two values are considered equal.</p>
</dd></dl>

<p>x is the point at which to compute the <a href="Jacobian.html"><code>Jacobian</code></a>.</p>

<p>fx is f.values(x).</p>

    </section>

      <section id="5Buntitled-5D" class="documentation-section">





                <section id="public-instance-5Buntitled-5D-method-details" class="method-section">
                <header>
                <h3>Public Instance Methods</h3>
                </header>

                  <div id="method-i-dfdxi" class="method-detail ">
                            <div class="method-heading">
                              <span class="method-name">dfdxi</span><span
                                class="method-args">(f,fx,x,i)</span>
                              <span class="method-click-advice">click to toggle source</span>
                            </div>

                            <div class="method-description">
                              <p>Computes the derivative of <code>f[i]</code> at <code>x[i]</code>. <code>fx</code> is the value of <code>f</code> at <code>x</code>.</p>

                              <div class="method-source-code" id="dfdxi-source">
            <pre><span class="ruby-comment"># File bigdecimal/lib/bigdecimal/jacobian.rb, line 47</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">dfdxi</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>,<span class="ruby-identifier">i</span>)
  <span class="ruby-identifier">nRetry</span> = <span class="ruby-value">0</span>
  <span class="ruby-identifier">n</span> = <span class="ruby-identifier">x</span>.<span class="ruby-identifier">size</span>
  <span class="ruby-identifier">xSave</span> = <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>]
  <span class="ruby-identifier">ok</span> = <span class="ruby-value">0</span>
  <span class="ruby-identifier">ratio</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span>
  <span class="ruby-identifier">dx</span> = <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>].<span class="ruby-identifier">abs</span><span class="ruby-operator">/</span><span class="ruby-identifier">ratio</span>
  <span class="ruby-identifier">dx</span> = <span class="ruby-identifier">fx</span>[<span class="ruby-identifier">i</span>].<span class="ruby-identifier">abs</span><span class="ruby-operator">/</span><span class="ruby-identifier">ratio</span> <span class="ruby-keyword">if</span> <span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">dx</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>)
  <span class="ruby-identifier">dx</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">one</span><span class="ruby-operator">/</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">ten</span>     <span class="ruby-keyword">if</span> <span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">dx</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>)
  <span class="ruby-keyword">until</span> <span class="ruby-identifier">ok</span><span class="ruby-operator">&gt;</span><span class="ruby-value">0</span> <span class="ruby-keyword">do</span>
    <span class="ruby-identifier">deriv</span> = []
    <span class="ruby-identifier">nRetry</span> <span class="ruby-operator">+=</span> <span class="ruby-value">1</span>
    <span class="ruby-keyword">if</span> <span class="ruby-identifier">nRetry</span> <span class="ruby-operator">&gt;</span> <span class="ruby-value">100</span>
      <span class="ruby-identifier">raise</span> <span class="ruby-string">&quot;Singular Jacobian matrix. No change at x[&quot;</span> <span class="ruby-operator">+</span> <span class="ruby-identifier">i</span>.<span class="ruby-identifier">to_s</span> <span class="ruby-operator">+</span> <span class="ruby-string">&quot;]&quot;</span>
    <span class="ruby-keyword">end</span>
    <span class="ruby-identifier">dx</span> = <span class="ruby-identifier">dx</span><span class="ruby-operator">*</span><span class="ruby-identifier">f</span>.<span class="ruby-identifier">two</span>
    <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>] <span class="ruby-operator">+=</span> <span class="ruby-identifier">dx</span>
    <span class="ruby-identifier">fxNew</span> = <span class="ruby-identifier">f</span>.<span class="ruby-identifier">values</span>(<span class="ruby-identifier">x</span>)
    <span class="ruby-keyword">for</span> <span class="ruby-identifier">j</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
      <span class="ruby-keyword">if</span> <span class="ruby-operator">!</span><span class="ruby-identifier">isEqual</span>(<span class="ruby-identifier">fxNew</span>[<span class="ruby-identifier">j</span>],<span class="ruby-identifier">fx</span>[<span class="ruby-identifier">j</span>],<span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>,<span class="ruby-identifier">f</span>.<span class="ruby-identifier">eps</span>) <span class="ruby-keyword">then</span>
        <span class="ruby-identifier">ok</span> <span class="ruby-operator">+=</span> <span class="ruby-value">1</span>
        <span class="ruby-identifier">deriv</span> <span class="ruby-operator">&lt;&lt;=</span> (<span class="ruby-identifier">fxNew</span>[<span class="ruby-identifier">j</span>]<span class="ruby-operator">-</span><span class="ruby-identifier">fx</span>[<span class="ruby-identifier">j</span>])<span class="ruby-operator">/</span><span class="ruby-identifier">dx</span>
      <span class="ruby-keyword">else</span>
        <span class="ruby-identifier">deriv</span> <span class="ruby-operator">&lt;&lt;=</span> <span class="ruby-identifier">f</span>.<span class="ruby-identifier">zero</span>
      <span class="ruby-keyword">end</span>
    <span class="ruby-keyword">end</span>
    <span class="ruby-identifier">x</span>[<span class="ruby-identifier">i</span>] = <span class="ruby-identifier">xSave</span>
  <span class="ruby-keyword">end</span>
  <span class="ruby-identifier">deriv</span>
<span class="ruby-keyword">end</span></pre>
                              </div>
                            </div>


                          </div>

                  <div id="method-i-isEqual" class="method-detail ">
                            <div class="method-heading">
                              <span class="method-name">isEqual</span><span
                                class="method-args">(a,b,zero=0.0,e=1.0e-8)</span>
                              <span class="method-click-advice">click to toggle source</span>
                            </div>

                            <div class="method-description">
                              <p>Determines the equality of two numbers by comparing to zero, or using the epsilon value</p>

                              <div class="method-source-code" id="isEqual-source">
            <pre><span class="ruby-comment"># File bigdecimal/lib/bigdecimal/jacobian.rb, line 30</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">isEqual</span>(<span class="ruby-identifier">a</span>,<span class="ruby-identifier">b</span>,<span class="ruby-identifier">zero</span>=<span class="ruby-value">0.0</span>,<span class="ruby-identifier">e</span>=<span class="ruby-value">1.0e-8</span>)
  <span class="ruby-identifier">aa</span> = <span class="ruby-identifier">a</span>.<span class="ruby-identifier">abs</span>
  <span class="ruby-identifier">bb</span> = <span class="ruby-identifier">b</span>.<span class="ruby-identifier">abs</span>
  <span class="ruby-keyword">if</span> <span class="ruby-identifier">aa</span> <span class="ruby-operator">==</span> <span class="ruby-identifier">zero</span> <span class="ruby-operator">&amp;&amp;</span>  <span class="ruby-identifier">bb</span> <span class="ruby-operator">==</span> <span class="ruby-identifier">zero</span> <span class="ruby-keyword">then</span>
    <span class="ruby-keyword">true</span>
  <span class="ruby-keyword">else</span>
    <span class="ruby-keyword">if</span> ((<span class="ruby-identifier">a</span><span class="ruby-operator">-</span><span class="ruby-identifier">b</span>)<span class="ruby-operator">/</span>(<span class="ruby-identifier">aa</span><span class="ruby-operator">+</span><span class="ruby-identifier">bb</span>)).<span class="ruby-identifier">abs</span> <span class="ruby-operator">&lt;</span> <span class="ruby-identifier">e</span> <span class="ruby-keyword">then</span>
      <span class="ruby-keyword">true</span>
    <span class="ruby-keyword">else</span>
      <span class="ruby-keyword">false</span>
    <span class="ruby-keyword">end</span>
  <span class="ruby-keyword">end</span>
<span class="ruby-keyword">end</span></pre>
                              </div>
                            </div>


                          </div>

                  <div id="method-i-jacobian" class="method-detail ">
                            <div class="method-heading">
                              <span class="method-name">jacobian</span><span
                                class="method-args">(f,fx,x)</span>
                              <span class="method-click-advice">click to toggle source</span>
                            </div>

                            <div class="method-description">
                              <p>Computes the <a href="Jacobian.html"><code>Jacobian</code></a> of <code>f</code> at <code>x</code>. <code>fx</code> is the value of <code>f</code> at <code>x</code>.</p>

                              <div class="method-source-code" id="jacobian-source">
            <pre><span class="ruby-comment"># File bigdecimal/lib/bigdecimal/jacobian.rb, line 79</span>
<span class="ruby-keyword">def</span> <span class="ruby-identifier ruby-title">jacobian</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>)
  <span class="ruby-identifier">n</span> = <span class="ruby-identifier">x</span>.<span class="ruby-identifier">size</span>
  <span class="ruby-identifier">dfdx</span> = <span class="ruby-constant">Array</span>.<span class="ruby-identifier">new</span>(<span class="ruby-identifier">n</span><span class="ruby-operator">*</span><span class="ruby-identifier">n</span>)
  <span class="ruby-keyword">for</span> <span class="ruby-identifier">i</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
    <span class="ruby-identifier">df</span> = <span class="ruby-identifier">dfdxi</span>(<span class="ruby-identifier">f</span>,<span class="ruby-identifier">fx</span>,<span class="ruby-identifier">x</span>,<span class="ruby-identifier">i</span>)
    <span class="ruby-keyword">for</span> <span class="ruby-identifier">j</span> <span class="ruby-keyword">in</span> <span class="ruby-value">0</span><span class="ruby-operator">...</span><span class="ruby-identifier">n</span> <span class="ruby-keyword">do</span>
      <span class="ruby-identifier">dfdx</span>[<span class="ruby-identifier">j</span><span class="ruby-operator">*</span><span class="ruby-identifier">n</span><span class="ruby-operator">+</span><span class="ruby-identifier">i</span>] = <span class="ruby-identifier">df</span>[<span class="ruby-identifier">j</span>]
    <span class="ruby-keyword">end</span>
  <span class="ruby-keyword">end</span>
  <span class="ruby-identifier">dfdx</span>
<span class="ruby-keyword">end</span></pre>
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